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- Hideyuki Ishi
--Analysis on regular convex cones associated to decomposable graphs
--Regular convex cones of positive definite real symmetric matrices with
prescribed zero entries
have been studied intensively in multivariate statistics. It turned
out that analysis on the cone is quite
feasible if the zero pattern corresponds to a decomposable graph.
Indeed, an explicit formula is known
for the Fourier-Laplace transform of a product of powers of minors
over the cone. Inspired by these statistic works,
we develop analysis on the cone in a similar way to theory of
homogeneous cones. In particular, we consider
Riesz distributions on the cone and associated b-functions.
-Yumiko Hironaka
--Spherical functions on certain $p$-adic homogeneous spaces, and some
relation to PV-theory
--First I want to introduce a typical spherical function on certain
homogeneous space $X$, and give its expression formula by using
functional equations of sph. f's and data of the group. In this talk
everything is assumed to be defined over a $\mathfrak{p}$-adic field $k$.
Typical sph. f's are obtained by Poisson transform from relative
$P$-invariant on $X$, where $P$ is a minimal parabolic that has Zarisky
open orbit in $X$ over $\overline{k}$, and their functional equations
are often reduced to those for certain limited type of prehomogeneous
vector spaces.
Then I want to discuss about some spaces of sesquiliear forms, and give
explicit formulas of sph.f's by using specialized Hall-Littlewood
polynomials associated to the root system, parametrization of all the
sph.f's, and Plancherel formula.
(The latter half is joint work with Y. Komori.)
-Koichi Takase
--On spherical functions of supercuspidal representations of $GL_n(F)$ and prehomogeneous vector spaces over finite fields
--We will consider the spherical function of a square-integrable irreducible representation of $G$ with respect to a $K$-type with a compact subgroup $K$ of $G$, and want to determine the non-zero set of the Fourier transform of the spherical function. In the case of the holomorphic discrete series, there exists a close relationship between the non-zero set and a pre homogeneous vector space of parabolic type associated with a boundary component of $G/K$. We can consider a parallel problem for $p$-adic reductive group. In this talk, I will report on the result of an experiment for the simplest case of an irreducible level-zero super cuspidal representation of $GL_n(F)$.
![[PukiWiki]](image/pukiwiki.png "[PukiWiki]")
